Crucial Electromagnetism Formulas

These notes cover essential electromagnetism formulas that connect electric and magnetic fields. Key concepts include Maxwell's Equations, Coulomb's Law, and Faraday's Law, forming a foundation for understanding classical electromagnetism and its applications in physics and engineering.

1. Maxwell's Equations (in differential and integral forms)

   • Describe the fundamental relationship between electric and magnetic fields.
   • Consist of four equations: Gauss's Law, Gauss's Law for Magnetism, Faraday's Law, and Ampère-Maxwell Law.
   • Provide a complete framework for classical electromagnetism, optics, and electric circuits.

2. Coulomb's Law

   • Defines the electrostatic force between two point charges.
   • The force is directly proportional to the product of the charges and inversely proportional to the square of the distance between them.
   • Fundamental for understanding electric fields and potentials.

3. Gauss's Law

   • Relates the electric flux through a closed surface to the charge enclosed by that surface.
   • Useful for calculating electric fields in symmetric charge distributions.
   • Integral form: ∮ E · dA = Q_enc/ε₀; differential form: ∇ · E = ρ/ε₀.

4. Ampère's Law

   • Relates the magnetic field around a closed loop to the electric current passing through the loop.
   • Integral form: ∮ B · dl = μ₀ I_enc; differential form: ∇ × B = μ₀ J + μ₀ε₀ ∂E/∂t.
   • Key for analyzing magnetic fields in circuits and inductors.

5. Faraday's Law of Induction

   • States that a changing magnetic field induces an electromotive force (EMF) in a closed loop.
   • Integral form: ∮ E · dl = -dΦ_B/dt; differential form: ∇ × E = -∂B/∂t.
   • Fundamental principle behind electric generators and transformers.

6. Biot-Savart Law

   • Describes the magnetic field generated by a steady electric current.
   • The magnetic field is proportional to the current and inversely proportional to the square of the distance from the current element.
   • Essential for calculating magnetic fields in complex current configurations.

7. Lorentz Force Law

   • Describes the force experienced by a charged particle moving in electric and magnetic fields.
   • F = q(E + v × B), where F is the force, q is the charge, E is the electric field, v is the velocity, and B is the magnetic field.
   • Crucial for understanding the motion of charged particles in electromagnetic fields.

8. Poynting Vector

   • Represents the directional energy flux (the rate of energy transfer per unit area) of an electromagnetic field.
   • Defined as S = E × H, where E is the electric field and H is the magnetic field.
   • Important for analyzing energy flow in electromagnetic systems.

9. Wave Equation for Electromagnetic Waves

   • Describes how electric and magnetic fields propagate through space.
   • Derived from Maxwell's equations, showing that electromagnetic waves travel at the speed of light.
   • Fundamental for understanding light and other electromagnetic radiation.

10. Larmor Formula for Radiating Charges

    • Gives the power radiated by a non-relativistic charged particle undergoing acceleration.
    • P = (2/3)(q²a²)/(c³), where P is the power, q is the charge, a is the acceleration, and c is the speed of light.
    • Key for understanding radiation from accelerating charges.

11. Electromagnetic Potential (scalar and vector potentials)

    • Scalar potential (φ) and vector potential (A) are used to describe electric and magnetic fields.
    • E = -∇φ - ∂A/∂t and B = ∇ × A.
    • Simplifies calculations in electromagnetism, especially in complex geometries.

12. Continuity Equation

    • Expresses the conservation of electric charge in a given volume.
    • ∂ρ/∂t + ∇ · J = 0, where ρ is charge density and J is current density.
    • Fundamental for understanding charge conservation in electromagnetic systems.

13. Ohm's Law

    • Relates the current flowing through a conductor to the voltage across it and its resistance.
    • V = IR, where V is voltage, I is current, and R is resistance.
    • Essential for circuit analysis and electrical engineering.

14. Polarization and Magnetization Equations

    • Describe how materials respond to electric and magnetic fields.
    • Polarization (P) relates to the electric field (E) as P = ε₀χE; magnetization (M) relates to the magnetic field (H) as M = χ_mH.
    • Important for understanding material properties in electromagnetism.

15. Boundary Conditions for Electromagnetic Fields

    • Define how electric and magnetic fields behave at the interface between different media.
    • Conditions include continuity of the tangential components of E and H, and the normal components of D and B.
    • Crucial for solving problems involving interfaces in optics and electromagnetism.